Laws of perfect gases

Assignment # 2

Laws of perfect gases:-

A Perfect gas (an Ideal gas) may be defined as a state of a substance, whose evaporation from its liquid state is complete, and strictly obeys all the gas laws under all conditions of temperature and pressure, In actual practice, there in no real or actual gas which strictly obeys the laws over the entire range of temperature and pressure, but the real gases which are ordinarily difficult to liquefy , such as oxygen, nitrogen , hydrogen and air , with in certain temperature and pressure limits, may be regarded as perfect gases. 

Laws of Perfect Gases:-

The physical properties of a gas are controlled by the fallowing their variables.

1- Pressure generated by the gas

2. Volume occupied by the gas

3. Temperature of the gas

The behavior of a perfect gas undergoing any change in the above mentioned variable is governed by the fallowing laws which have been established from experimental results. 

Boyle’s Law

Charles Law

Gay-Lussac Law

Boyle’s Law:-

This law was formulated by Robert Boyle in 1662. It states “the absolute pressure of a given mass of a perfect gas varies inversely as its volume with in the temperature remains constant”.

P ∝ 1/V

PV = constant

At different states, more useful form of the above equation is 

P1V1 = P2V2 = P3V3-----------------= Constant

Suffixes 1,2 and 3------ refer to different sets of conditions. 

Charle’s Law

This law was formulated by a French man Jacques A.C charle’s in about 1787. It may be stated in the fallowing two different forms.

The volume of a given mass of a perfect gas varies directly as into absolute temperature, when the absolute pressure remains constant.

V ∝  V/T = Constant

V1/T1 = V2/T2 = V3/T3 ------------------- Constant

All the perfect gases change in volume by 1/273 th of its original volume at O℃ for every 1℃ change in temperature, when the pressure remains constant. 

Vo = Volume of a given mass of gas at O℃ and

Vt = Volume of the same mass of gas at t℃.

T= Absolute temperature corresponding to t℃.

To = Absolute temperature corresponding to O℃.

A little consideration will show, that the volume of a gas goes on decreasing by 1/273 th of its original volume for every 1℃ decrease in temperature, it is thus obvious, that at a temperature of -273℃, the volume of the gas would become zero, the temperature at which the volume of a gas becomes zero is called absolute zero temperature.

Gay-Lussac Law:-

This law states that absolute pressure of a given mass of a perfect gas varies directly as its absolute temperature, when the volume remains constant. 

P ∝ T or P/T = constant

P1/T1 = P2/T2  = P3/T3  -----------Constant 

General Gas Equation

In the previous section we have discussed the gas lows which give us the relation between the two variables when the third variable is constant, but in actual practice all the three variables pressure, volume and temperature, change simultaneously, in order to deal with a practical cases, the boyle’s law and charle’s law are combined together, which give us a general gas equation.

According to Boyle’s Law

P ∝  1/V or V ∝  1/P 

According to Charle’s Law

V ∝ T

It is thus obvious that 

V ∝ I/P and T both

V ∝ T/P

PV  ∝ T or PV = CT

Where C is a constant, whose value depend upon the mass and properties of the gas concerned the more useful form of the general gas equation is 

P1V1/T1 = P2V2/T2 = P3V3/T3  -------------- Constant

Example

A gas occupies a volume of 0.1m^3 at temperature of 20 ℃ and a pressure of 1.5 bar. Find the final temperature of the gas, if it is compressed to a pressure of 7.5 bar and occupies a volume of 0.04 m^3.

Solution

V1 = 0.1 m^3

T1 = 20 ℃ = 20 + 273 = 293K

V2 = 0.04 m^3

T2 = ? find temperature 

P1 = 1.5 bar = 1.5 x 〖10〗^5 K or 

0.15 x 〖10〗^6 K

P2 = 7.5 bar = 7.5 x 〖10〗^5 K or 

0.75 x 〖10〗^6 K

We know the general gas equation

P1V1/T1 = P2V2/T2

T2 = (P2V2 T1)/P1T2

T2 = (7.5 x 〖10〗^5   x 0.04 x 293)/(1.5 x 〖10〗^5  x 0.1)

T2 = 586 K

    = 586 – 273 = 313℃

Joule’s Law

The change of the internal energy of a perfect gas is directly proportional to the change of temperature.

dE ∝ dT or dE = mc dT

m = mass of a gas

C = Constant of proportionality know as specific heat. 

dE = mc ( T2-T1 )

an important aspect of this law is that if the temperature of a given mass m of a gas changes from T1 to T2  then the internal energy ( E2-E1) will be same irrespective,  of the manner how the pressure ( P ) and volume ( V ) of the gas have changed.